MODULAR ARITHMETIC WORKSHEET

1.  Find the remainders when 70004 and 778 is divided by 7.

2.  Determine the value of d such that 15 ≡ 3 (mod d).

3.  Find the least positive value of x such that

67 + x ≡ 1 (mod 4)

4.  Find the least positive value of x such that

98 ≡ (x + 4) (mod 5)

5.  Solve 8x ≡ 1 (mod 11).

6.  Compute x, such that  10⁴ ≡ x (mod 19).

7.  Find the number of integer solutions of

3x ≡ 1 (mod 15)

8.  What month is 19 months after July ?

9. A man starts his journey from place X to place Y by train. He starts at 22.30 hours on Wednesday. If it takes 32 hours of travelling time and assuming that the train is not late, when will he reach the place Y?

10. Kala and Vani are friends. Kala says, “Today is my birthday” and she asks Vani, “When will you celebrate your birthday?” Vani replies, “Today is Monday and I celebrated my birthday 75 days ago”. Find the day when Vani celebrated her birthday.

1. Answer :

Since 70000 is divisible by 7 

70000 ≡ 0 (mod 7)

70000 + 4 ≡ 0 + 4 (mod 7)

70004 ≡ 4 (mod 7)

Therefore, the remainder when 70004 is divided by 7 is 4.

Since 777 is divisible by 7

777 ≡ 0 (mod 7)

777 + 1 ≡ 0 + 1 (mod 7)

778 ≡ 1 (mod 7)

Therefore, the remainder when 778 is divided by 7 is 1.

2. Answer :

15 ≡ 3 (mod d) means 15 - 3 = kd, for some integer k.

12 = kd

gives d divides 12.

The divisors of 12 are 1, 2, 3, 4, 6, 12. But d should be larger than 3 and so the possible values for d are 4, 6, 12.

3. Answer :

67 + x ≡ 1 (mod 4)

67 + x - 1 = 4n , for some integer n

66 + x = 4n

(66 + x) is a multiple of 4.

Therefore, the least positive value of x must be 2, since 68 is the nearest multiple of 4 more than 66.

4. Answer :

98 ≡ (x + 4) (mod 5)

98 - (x + 4) = 5n , for some integer n

98 - x - 4 = 5n

94 - x = 5n

(94 - x) is a multiple of 5.

Therefore, the least positive value of x must be 4.

Since 94 - 4 = 90 is the nearest multiple of 5 less than 94.

5. Answer : 

8x ≡ 1 (mod 11) can be written as 8x - 1 = 11k, for some integer k.

x = (11k + 1)/8

When we put k = 5, 13, 21, 29,... then (11k + 1) is divisible by 8.

x = [11(5) + 1]/8 = 7

x = [11(13) + 1]/8 = 18

Therefore, the solutions are 7, 18, 29, 40, …

6. Answer : 

10² = 100 ≡ 5 (mod 19)

10⁴ = (10²)² ≡ 5² (mod 19)

10⁴ ≡ 25 (mod 19)

Since 25 ≡ 6 (mod 19),

10⁴ ≡ 6 (mod 19)

Therefore, x = 6. 

7. Answer : 

3x ≡ 1 (mod 15) can be written as

3x - 1 = 15k for some integer k

3x = 15k +1

x = (15k + 1)/3

x = 5k + 1/3

Since 5k is an integer, 5k + 1/3 cannot be an integer. 

So there is no integer solution.

8. Answer : 

July corresponds to 7 in month arithmetic. 

We have to find the month which is 19 months after July. 

7 + 19 (mod 12) ≡ 26 (mod 12) ≡ 2 (mod 12)

The month for the number 2 is February.

So, the month which is after 19 months will be February .

9. Answer : 

Starting time 22.30, Travelling time 32 hours. Here we use modulo 24.

The reaching time is

22.30 + 32 (mod 24) ≡ 54.30 (mod 24)

≡ 6.30 (mod24)

Thus, he will reach the place Y on Friday at 6.30 hours.

Answer 10 :

Let us associate the numbers 0, 1, 2, 3, 4, 5, 6 to represent the weekdays from Sunday to Saturday respectively.

Vani says today is Monday. So the number for Monday is 1. Since Vani’s birthday was 75 days ago, we have to subtract 75 from 1 and take the modulo 7, since a week contain 7 days.

–74 (mod 7) ≡ -4 (mod 7) ≡ 7 - 4 (mod 7) ≡ 3 (mod 7)

(Since -74 – 3 = -77 is divisible by 7)

Thus,

1 - 75 ≡ 3 (mod 7)

The day for the number 3 is Wednesday.

Therefore, Vani’s birthday must be on Wednesday.

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